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Full text of "Treatise On Analysis Vol-Ii"

6    LOCALLY FINITE COVERINGS AND PARTITIONS OF UNITY       21

Put Kn = 0W — Vn,l9 which is a compact set for all n g: 0 (we put Un = 0 for
all n < 0). For each n *> 0, the open set Vn + l — 0^-2 is a neighborhood of
Kn. Hence, for each x e Kn, there exists an open neighborhood WŁn) € 93 of jc,
contained in U^+x — UM^.2 a°d contained in one of the sets A^. There exists a
finite number of points xt e Kn (1 <Ł / <; /?„) such that the sets W^} cover Kn.
Arrange the sets W^l) (m ^ 0, 1 g / <; /?,„ for each m) in a sequence in any
way, and let (Bn) denote this sequence. It is clear that (Bn) is an open covering
of E which is finer than (AA)A 6 L, and that the sets Bn are relatively compact.
Thus all that has to be checked is that the covering (Bn) is locally finite.
Let z be any point of E, and let n be the least integer such that z e UM. Since
z$Un-it there exists a neighborhood T of z contained in Un and not
meeting 0M_2. Consequently T can intersect only the sets W^ for which
n — 2^m^n+ 1, and the number of these sets is finite.

(12.6.2) Let (An) be a denumerable locally finite open covering of a metrizable
space E. Then there exists an open covering (BJ ofE such that Bn c Anfor alln.

We shall define the family (BM) by induction on n, in such a way that
Bn c An for each n, and such that for each n the family consisting of the Bfc
with k^n and the Ay- withy > /? is an open covering of E. Suppose that the
sets Bn have been defined for n < m. Then the Bn with n < m and the Ay with
j ^m cover E. Let C be the open set which is the union of the Brt with n < m
and the Aj withy > m -f 1, so that we have E — Am c C. We shall show that
there exists an open set V such that E — Am c= V c V c C. If E = Am,
we may take V = 0. If C = E, we may take V = E. If neither E — Am nor
E — C is empty, there exists a continuous mapping / of E into the interval
[0, 1] of R, which is equal to 0 on E - Am and to 1 on E - C (4.5.2). We then
take V to be the open set of points y such that/(j/) < Ł, and then V is con-
tained in the closed set of points y such that f(y) ^ ^, and hence V c C.
Put Bw = E - V. Then we have Bm <= E - V c Am and B,,, u C = E, so that
the sets Bnwith n ^ m satisfy the required conditions, and the induction can
proceed. For each x e E, there exists an integer n such that x $ Am for all
m > n, and therefore x e Bk for some k ^ n. Hence the sets BM cover E.

It is clear that the same argument will apply if the covering (An) is finite.

If E is a topological space and/is a mapping of E into a real vector space
F or into K, the support off (denoted by Supp(/)) is defined to be the smallest
closed set S in E such that/vanishes on E — S. In other words, Supp(/) is the
closure in E of the set of points x e E such that/(x) ^ 0; or again it is the set
of points x e E with the property that every neighborhood of x in E contains a
point y such that/0/) ^ 0.a nonempty open interval in R, there exists no nonconstant mapping